Impossible Tuning Made Possible: A New Expert Algorithm and Its Applications
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We resolve the long-standing "impossible tuning" issue for the classic expert problem and show that, it is in fact possible to achieve regret O(√((ln d)∑_t ℓ_t,i^2)) simultaneously for all expert i in a T-round d-expert problem where ℓ_t,i is the loss for expert i in round t. Our algorithm is based on the Mirror Descent framework with a correction term and a weighted entropy regularizer. While natural, the algorithm has not been studied before and requires a careful analysis. We also generalize the bound to O(√((ln d)∑_t (ℓ_t,i-m_t,i)^2)) for any prediction vector m_t that the learner receives, and recover or improve many existing results by choosing different m_t. Furthermore, we use the same framework to create a master algorithm that combines a set of base algorithms and learns the best one with little overhead. The new guarantee of our master allows us to derive many new results for both the expert problem and more generally Online Linear Optimization.
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